

A316833


Sums of four distinct odd squares.


3



84, 116, 140, 156, 164, 180, 196, 204, 212, 228, 236, 244, 252, 260, 276, 284, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548, 556, 564, 572, 580, 588, 596, 604, 612, 620, 628, 636, 644, 652, 660
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OFFSET

1,1


COMMENTS

Theorem (Conjectured by R. William Gosper, proved by M. D. Hirschhorn): Any sum of four distinct odd squares is the sum of four distinct even squares.
The proof uses the following identity:
(4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (abc+d)^2 + (ab+cd)^2 + (a+bcd)^2 ].
All terms == 4 (mod 8). Are all numbers == 4 (mod 8) and > 412 members of the sequence?  Robert Israel, Jul 20 2018


REFERENCES

R. William Gosper and Stephen K. Lucas, Postings to Math Fun Mailing List, July 19 2018
Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000


MAPLE

N:= 1000: # to get all terms <= N
V:= Vector(N):
for a from 1 to floor(sqrt(N/4)) by 2 do
for b from a+2 to floor(sqrt((Na^2)/3)) by 2 do
for c from b+2 to floor(sqrt((Na^2b^2)/2)) by 2 do
for d from c + 2 by 2 do
r:= a^2+b^2+c^2+d^2;
if r > N then break fi;
V[r]:= V[r]+1
od od od od:
select(t > V[t]>=1, [$1..N]); # Robert Israel, Jul 20 2018


CROSSREFS

A316834 lists the subsequence for which the representation is unique.
Cf. A004433, A316835.
Sequence in context: A157119 A209204 A219801 * A316834 A227734 A192322
Adjacent sequences: A316830 A316831 A316832 * A316834 A316835 A316836


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jul 19 2018


STATUS

approved



